3 I changed my initial statement to reflect the comment of David.; added 2 characters in body

Mordell gives partial integer solutions to a problem for which

I would like to find all integer triples (x,y,z) such that: $\prod_{\theta}(x + y \theta + z \theta^2)=1$, where $\theta$ runs through the general solutionsolutions to the cubic $x^3 + x^2 - 2x - 1=0$.

In his book "Diophantine equations" (p. 111-12) Mordell gives the equation

$w^n = \prod_{\theta}(x + y \theta + z \theta^2)$

where $\theta=\theta_1, \theta_2, \theta_3$ are the solutions to a cubic equation with integer coefficients.

Mordell's partial solution is,

$x + y\theta_1 + z \theta_1^2 = (p + q\theta_1 + r\theta_1^2)^n$,

$x + y\theta_2 + z \theta_2^2 = (p + q\theta_2 + r\theta_2^2)^n$,

$x + y\theta_3 + z \theta_3^2 = (p + q\theta_3 + r\theta_3^2)^n$,

$w = \prod_{\theta}(p + q \theta + r \theta^2)$

where $p,q,r$ are arbitrary integers and $n$ runs through the integers.''

He continues to say, " the general solution depends upon the theory of algebraic numbers and is connected with the units in an algebraic number field"

I am in the example where $w=1$ and $\theta_i$ are the solutions to the polynomial $x^3 + x^2 - 2x - 1=0$field".

2 fixed typo: changed indecies on the left hand side of particular solutions.

Mordell gives partial integer solutions to a problem for which I would like the general solution.

In his book "Diophantine equations" (p. 111-12) Mordell gives the equation

$w^n = \prod_{\theta}(x + y \theta + z \theta^2)$

where $\theta=\theta_1, \theta_2, \theta_3$ are the solutions to a cubic equation with integer coefficients.

Mordell's partial solution is,

$x + y\theta_1 + z \theta_1^2 = (p + q\theta_1 + r\theta_1^2)^n$,

$x + y\theta_1 y\theta_2 + z \theta_1^2 theta_2^2 = (p + q\theta_2 + r\theta_2^2)^n$,

$x + y\theta_1 y\theta_3 + z \theta_1^2 theta_3^2 = (p + q\theta_3 + r\theta_3^2)^n$,

$w = \prod_{\theta}(p + q \theta + r \theta^2)$

where $p,q,r$ are arbitrary integers and $n$ runs through the integers.''

He continues to say, " the general solution depends upon the theory of algebraic numbers and is connected with the units in an algebraic number field"

I am in the example where $w=1$ and $\theta_i$ are the solutions to the polynomial $x^3 + x^2 - 2x - 1=0$.

1

# representation of integers as the product of linear forms in three variables

Mordell gives partial integer solutions to a problem for which I would like the general solution.

In his book "Diophantine equations" (p. 111-12) Mordell gives the equation

$w^n = \prod_{\theta}(x + y \theta + z \theta^2)$

where $\theta=\theta_1, \theta_2, \theta_3$ are the solutions to a cubic equation with integer coefficients.

Mordell's partial solution is,

$x + y\theta_1 + z \theta_1^2 = (p + q\theta_1 + r\theta_1^2)^n$,

$x + y\theta_1 + z \theta_1^2 = (p + q\theta_2 + r\theta_2^2)^n$,

$x + y\theta_1 + z \theta_1^2 = (p + q\theta_3 + r\theta_3^2)^n$,

$w = \prod_{\theta}(p + q \theta + r \theta^2)$

where $p,q,r$ are arbitrary integers and $n$ runs through the integers.''

He continues to say, " the general solution depends upon the theory of algebraic numbers and is connected with the units in an algebraic number field"

I am in the example where $w=1$ and $\theta_i$ are the solutions to the polynomial $x^3 + x^2 - 2x - 1=0$.